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Why Is Hume's Principle Not Good Enough for Frege?

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Why Is Hume's Principle Not Good Enough for Frege? Why is Hume's Principle not good enough for Frege? Sicun Gao January 5, 2011 1 Introduction Unfair as it is, the most well-known property of Frege's foundation of arithmetic seems to be \inconsistent". His attempt was to interpret arithmetic in second- order logic with the following \Basic Law V": rF = rG $ 8x(F (x) = G(x))(1) where rF denotes the extension of a concept (equivalent to its graph). This axiom expresses that two concepts have the same extension iff they assign the same evaluations for all the entities. Russell discovered that this axiom, together with the comprehension principle (which asserts that every formula corresponds to some predicate) in second-order logic, leads to inconsistency (2) 9X(rX 2 X $ rX 62 X): This paradox is regarded as the deathblow for Frege's system. However, in recent years it has been realized that there are various ways of salvaging Frege's system from such inconsistency. Frege's arithmetic can be shown consistent if any of the following weakening is applied: (i) First-order Frege arithmetic, whose only nonlogical vocabulary is the ex- tension operator r, and whose axioms are all first-order instances of Basic Law V is shown to be consistent [2]. (ii) If the concept in the comprehension principle (see Section 2) is restricted to a certain form, the system can become equi-interpretable with various subsystems of second-order arithmetic [11]. (iii) If the Basic Law V is substituted for the so-called Hume's Principle, which expresses that two concepts have the same cardinality if there exists a bijection between them, then the resulted system is equi-interpretable with full second-order arithmetic. The third direction retains the most of Frege's original system, and has drawn most philosophical attention. Most notably, it inspires the so-called neo-logicism, which aims to resurrect Frege's logicism program on the base of Hume's Principle. Neo-logicists hold the position that since arithmetic can be deduced from second-order logic and Hume's Principles alone, and it is possible 1 to build \a form of epistemic foundationalism in which logic is intended to play a foundational role in resolving specific epistemic challenges, such as our knowledge of arithmetic and analysis." [4] Interestingly, Frege himself had the chance of being a neo-logicist. If we take a close examination of Frege's original program of interpreting arithmetic, it was divided in two steps: (i) Reduce arithmetic to second-order logic plus Hume's principle, as Frege wrote in x87 of [7]: I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgements and conse- quently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. (ii) Further reduce Hume's Principle to more basic laws, as in x0 pf [6]: In my Grundlagen der Arithmetik, I sought to make it plausible that arithmetic is a branch of logic and need not borrow any ground of proof whatever from either experience or intuition. In the present book, this shall be confirmed, by the derivation of the simplest laws of Numbers by logical means alone. Given such a clear separation of the two steps, it is reasonable to imagine that Frege had the option of claiming that he has reduced arithmetic to a few very basic principles: Although the second step of the program was not completed, he could claim second-order logic plus Hume's Principle to be the foundation of arithmetic, as is now suggested by the neo-logicists. Richard Heck in [9] explicitly asked this question as follows (after showing Frege's original proofs of the Dedekind-Peano axioms from Hume's Principle). All of this having been said, the question arises why, upon receiving Russell's famous letter, Frege did not simply drop Axiom V, install Hume's Principle as an axiom, and claim himself to have established logicism anyway. What I do know is that the questions lately raised need answers: for, until we have such answers, we shall not understand the significance that Axiom V had for Frege, since we shall not understand why he could not abandon it in favor of Hume's Principle. That is to say, we shall not understand how he conceived the logicist project. This is the question I aim to investigate. This question can have a short answer: Russell's paradox put something very fundamental in Frege's belief into question, which can hardly be circumvented by a technical solution or partial claim. Frege's bewilderment about how to rethink about his logicism program is best reflected in his letter to Russell [5]: Your discovery of the contradiction has surprised me beyond words and, I should almost like to say, left me thunderstruck, because it has rocked the ground on which I meant to build arithmetic...It is all the more serious as the collapse of my law V seems to undermine not only 2 the foundations of my arithmetic but the only possible foundations of arithmetic as such... The question is, how do we apprehend logical objects? And I have found no other answer to it than this, We apprehend them as extensions of concepts, or more generally, as value-ranges of functions. I have always been aware that there were difficulties with this, and your discovery of the contradiction has added to them; but what other way is there? The fact that the underlying notions themselves can give rise to false knowl- edge baffled Frege. The notions of concepts, functions and their course-of- values are regarded as self-evident to Frege, about which he holds a rationalist's view. He regarded them as \the only possible foundation of arithmetic as such". Frege's commitment to the existence of the third realm, and more importantly our epistemological competence of grasping the realm, is fundamental to his program. Russell's paradox shows that the very foundation of this program is shaken because the properties of the logical objects, if they exist at all, can be even more puzzling than arithmetic itself. This very fact is disturbing in a much deeper sense for Frege, beyond the failure of deriving arithmetic per se. Along these lines, I aim to understand the reason why Frege regarded Hume's Principle as not fundamental enough, while Basic Law V is, had it been correct. To do this, we need to first understand Frege's plan of using Hume's Principle to deduce the Dedekind-Peano Axioms. Then, I look at Frege's own objection to relying on Hume's Principle as a basic law, to which he raise the famous Julius Caesar Problem. By examining Frege's view of this problem, I aim to clarify Frege's implicit commitment on both ontological and epistemological levels. On the other hand, the fact that Frege choosed to accept Law V instead, putting him under the colored-glasses of ours, shows his naivety about our com- petence in \grasping"the logical objects. I compare our contemporary skeptic attitude towards finding any analytic foundation for arithmetic with Frege's op- timism, and conclude that this shows the valuable progress we have achieved in our understanding of the notion of analyticity through logic and analytic philosophy. 2 Frege's Theorem Frege's theorem, as termed by George Boolos, states that the Dedekind-Peano Axioms can be interpreted in second-order logic with Hume's Principle as the only non-logical axiom. The proof of the theorem has been extracted from Frege's original works and nicely presented in [9, 13]. In this section, I outline the basic constructions in the proof to facilitate later discussions. We assume a standard system of second-order logic. The only non-logical axiom that is needed in Frege's arithmetic is the Hume's Principle, which we now formally state. Definition 2.1 (Cardinality Operator). Let P be any predicate. The cardinality of P is denoted as #P . Definition 2.2 (Hume's Principle). The cardinality operator is interpreted by the Hume's principle: (3) #P = #Q $ P ' Q 3 where (4) P ' Q = 9R(8x(P x ! 9!y(Qy ^ Rxy) ^ 8x(Qx ! 9!y(P y ^ Ryx)))) . This condition shows that there exists a bijection between P and Q. Naturally, cardinal numbers can be defined using existentially quantification: Definition 2.3 (Cardinal numbers). (5) Nx $ 9P (x = #P ) A main rule in Frege's system is the substitution rule, which asserts that any atomic formula can be substituted by a complex formula. It leads to the following so-called comprehension principle, which is important in the definition of numbers: Definition 2.4 (Comprehension Principle). (6) 9P 8x(P x $ '(x)) where '(x) does not contain free occurrence of P or x. (Here x can be a vector of n variables to cover the case of n-ary concepts.) It is easy to see how this principle is derived: 8x(F x $ F x) (Axiom)(7) 9G8x(Gx $ F x) (Existential Quantification)(8) 9G8x(Gx $ '(x)) (Substitution)(9) Note that the step 8 suggested that a strong existential commitment (first suggested by Boolos in [1]). We will return back to this point in Section 4. The principle says that for any formula '(x), there is a corresponding con- cept P . This allows us to introduce the λ expressions as a familiar notation. Definition 2.5 (λ-conversion). For any formula '(x), λϕ(x) is a predicate that satisfies: (10) 8y ((λx.ϕ(x))y $ '[y=x]) where '[y=x] denotes the usual substitution of x by y.
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